Timeline of numerical analysis after 1945

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The following is a timeline of numerical analysis after 1945, and deals with developments after the invention of the modern electronic computer, which began during Second World War. For a fuller history of the subject before this period, see timeline and history of mathematics.

## Contents

* 1 1940s
* 2 1950s
* 3 1960s
* 4 1970s
* 5 1980s
* 6 See also
* 7 References
* 8 Further reading
* 9 External links

## 1940s

* Monte Carlo simulation (voted one of the top 10 algorithms of the 20th century) invented at Los Alamos by von Neumann, Ulam and Metropolis.[1][2][3]
* Crank–Nicolson method was developed by Crank and Nicolson.[4]
* Dantzig introduces the simplex method (voted one of the top 10 algorithms of the 20th century) in 1947.[5]
* Turing formulated the LU decomposition method.[6]

## 1950s

* Successive over-relaxation was devised simultaneously by D.M. Young Jr.[7] and by H. Frankel in 1950.
* Hestenes, Stiefel, and Lanczos, all from the Institute for Numerical Analysis at the National Bureau of Standards, initiate the development of Krylov subspace iteration methods.[8][9][10][11] Voted one of the top 10 algorithms of the 20th century.
* Equations of State Calculations by Fast Computing Machines introduces the Metropolis–Hastings algorithm.[12]
* In numerical differential equations, Lax and Friedrichs invent the Lax-Friedrichs method.[13][14]
* Householder invents his eponymous matrices and transformation method (voted one of the top 10 algorithms of the 20th century).[15]
* Romberg integration[16]
* John G.F. Francis[17] and Vera Kublanovskaya[18] invent QR factorization (voted one of the top 10 algorithms of the 20th century).

## 1960s

* First recorded use of the term "finite element method" by Ray Clough,[19] to describe the methods of Courant, Hrenikoff, Galerkin and Zienkiewicz, among others. See also here.
* Exponential integration by Certaine and Pope.
* In computational fluid dynamics and numerical differential equations, Lax and Wendroff invent the Lax-Wendroff method.[20]
* Fast Fourier Transform (voted one of the top 10 algorithms of the 20th century) invented by Cooley and Tukey.[21]
* First edition of Handbook of Mathematical Functions by Abramowitz and Stegun, both of the U.S.National Bureau of Standards.[22]
* Broyden does new quasi-Newton method for finding roots in 1965.
* The MacCormack method, for the numerical solution of hyperbolic partial differential equations in computational fluid dynamics, is introduced by MacCormack in 1969.[23]
* Verlet (re)discovers a numerical integration algorithm, (first used in 1791 by Delambre, by Cowell and Crommelin in 1909, and by Carl Fredrik Störmer in 1907, hence the alternative names Störmer's method or the Verlet-Störmer method) for dynamics.

## 1970s

Creation of LINPACK and associated benchmark by Dongarra et al.,[24][25] as well as BLAS.

## 1980s

* Progress in wavelet theory throughout the decade, led by Daubechies et al.
* Creation of MINPACK.
* Fast multipole method (voted one of the top 10 algorithms of the 20th century) invented by Rokhlin and Greengard.[26][27][28]
* First edition of Numerical Recipes by Press, Teukolsky, et al.[29]
* In numerical linear algebra, the GMRES algorithm invented in 1986.[30]

## See also

* Scientific computing
* History of numerical solution of differential equations using computers
* Numerical analysis
* Timeline of computational mathematics

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## References

1. ↑ Metropolis, N. (1987). "The Beginning of the Monte Carlo method". Los Alamos Science 15: 125. http://library.lanl.gov/cgi-bin/getfile?15-12.pdf. . Accessed 5 may 2012.
2. ↑ S. Ulam, R. D. Richtmyer, and J. von Neumann (1947). Statistical methods in neutron diffusion. Los Alamos Scientific Laboratory report LAMS–551.
3. ↑ Metropolis, N.; Ulam, S. (1949). "The Monte Carlo method". Journal of the American Statistical Association 44 (247): 335–341. doi:10.1080/01621459.1949.10483310. PMID 18139350.
4. ↑ Crank, J. (John); Nicolson, P. (Phyllis) (1947). "A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type". Proc. Camb. Phil. Soc. 43 (1): 50-67. doi:10.1007/BF02127704.
5. ↑ "SIAM News, November 1994.". http://www.stanford.edu/group/SOL/dantzig.html. Hosted at Systems Optimization Laboratory, Stanford University, Huang Engineering Center .
6. ↑ A. M. Turing, Rounding-off errors in matrix processes. Quart. J Mech. Appl. Math. 1 (1948), 287–308 (according to Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Canada: Thomson Brooks/Cole, ISBN 0-534-99845-3.) .
7. ↑ Iterative methods for solving partial difference equations of elliptical type, PhD thesis, Harvard University, 1 May 1950, http://www.ma.utexas.edu/CNA/DMY/david_young_thesis.pdf, retrieved 15 June 2009
8. ↑ Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, J. Res. Natl. Bur. Stand. 49, 409–436 (1952).
9. ↑ Eduard Stiefel, U¨ ber einige Methoden der Relaxationsrechnung (in German), Z. Angew. Math. Phys. 3, 1–33 (1952).
10. ↑ Cornelius Lanczos, Solution of Systems of Linear Equations by Minimized Iterations, J. Res. Natl. Bur. Stand. 49, 33–53 (1952).
11. ↑ Cornelius Lanczos, An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, J. Res. Natl. Bur. Stand. 45, 255–282 (1950).
12. ↑ Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. (1953). "Equation of State Calculations by Fast Computing Machines". Journal of Chemical Physics 21 (6): 1087–1092. doi:10.1063/1.1699114. Bibcode: 1953JChPh..21.1087M.
13. ↑ Lax, PD (1954). "Weak solutions of nonlinear hyperbolic equations and their numerical approximation". Comm. Pure Appl. Math. 7: 159–193. doi:10.1002/cpa.3160070112.
14. ↑ Friedrichs, KO (1954). "Symmetric hyperbolic linear differential equations". Comm. Pure Appl. Math. 7 (2): 345–392. doi:10.1002/cpa.3160070206. Bibcode: 1954CPAM....7..345F.
15. ↑ Householder, A. S. (1958). "Unitary Triangularization of a Nonsymmetric Matrix". Journal of the ACM 5 (4): 339-342. doi:10.1145/320941.320947. https://hal.archives-ouvertes.fr/hal-01316095/file/p339householderb.pdf.
16. ↑ 1955
17. ↑ J.G.F. Francis, "The QR Transformation, I", The Computer Journal, 4(3), pages 265–271 (1961, received October 1959) online at oxfordjournals.org;J.G.F. Francis, "The QR Transformation, II" The Computer Journal, 4(4), pages 332–345 (1962) online at oxfordjournals.org.
18. ↑ Vera N. Kublanovskaya (1961), "On some algorithms for the solution of the complete eigenvalue problem," USSR Computational Mathematics and Mathematical Physics, 1(3), pages 637–657 (1963, received Feb 1961). Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki [Journal of Computational Mathematics and Mathematical Physics], 1(4), pages 555–570 (1961).
19. ↑ RW Clough, "The Finite Element Method in Plane Stress Analysis", Proceedings of 2nd ASCE Conference on Electronic Computation, Pittsburgh, PA, 8, 9 Sept. 1960.
20. ↑ P.D Lax; B. Wendroff (1960). "Systems of conservation laws". Commun. Pure Appl. Math. 13 (2): 217–237. doi:10.1002/cpa.3160130205. http://www.dtic.mil/get-tr-doc/pdf?AD=ADA385056.
21. ↑ Cooley, James W.; Tukey, John W. (1965). "An algorithm for the machine calculation of complex Fourier series". Math. Comput. 19 (90): 297–301. doi:10.1090/s0025-5718-1965-0178586-1. Bibcode: 1965MaCom..19..297C. http://attach3.bdwm.net/attach/0Announce/groups/GROUP_3/MathTools/D6714701A/D69595345/M.1089260001.A/CooleyJ_AlgMCC.pdf.
22. ↑ M Abramowitz and I Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Publisher: Dover Publications. Publication date: 1964; ISBN 0-486-61272-4;OCLC Number:18003605 .
23. ↑ MacCormack, R. W., The Effect of viscosity in hypervelocity impact cratering, AIAA Paper, 69-354 (1969).
24. ↑ J. Bunch; G. W. Stewart.; Cleve Moler; Jack J. Dongarra (1979). LINPACK User's Guide. Philadelphia, PA: SIAM.
25. ↑ The LINPACK Benchmark: Past, Present, and Future. Jack J. Dongarra, Piotr Luszczeky, and Antoine Petitetz. December 2001.
26. ↑ L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT, Cambridge, (1987).
27. ↑ Rokhlin, Vladimir (1985). "Rapid Solution of Integral Equations of Classic Potential Theory." J. Computational Physics Vol. 60, pp. 187–207.
28. ↑ Greengard, L.; Rokhlin, V. (1987). "A fast algorithm for particle simulations". J. Comput. Phys. 73 (2): 325–348. doi:10.1016/0021-9991(87)90140-9. Bibcode: 1987JCoPh..73..325G.
29. ↑ Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1986). Numerical Recipes: The Art of Scientific Computing. New York: Cambridge University Press. ISBN 0-521-30811-9.
30. ↑ Saad, Y.; Schultz, M.H. (1986). "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems". SIAM J. Sci. Stat. Comput. 7 (3): 856–869. doi:10.1137/0907058.

## Further reading

* "Top 10 Algorithms of the 20th Century". SIAM News. Society for Industrial and Applied Mathematics (SIAM). 2000. http://www.siam.org/news/news.php?id=637.

## External links

* The History of Numerical Analysis and Scientific Computing @ SIAM (Society for Industrial and Applied Mathematics)
* Ruttimann, Jacqueline (2006). "2020 computing: Milestones in scientific computing". Nature 440 (7083): 399–405. doi:10.1038/440399a. PMID 16554772. Bibcode: 2006Natur.440..399R.
* The Monte Carlo Method: Classic Papers
* Monte Carlo Landmark Papers
* “Must read” papers in numerical analysis. Discussion at MathOverflow based upon a selected reading list on Lloyd N. Trefethen's personal site.

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